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JOURNAL OF COMBINATORIAL THEORY, Series B 42, 313-318 (1987)

Coloring Perfect (K, - e) - Free Graphs

ALAN TUCKER *

Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, New York 11794

Communicated by the Managing Editors Received June 25. 1984

This note proves the Strong Perfect Graph Conjecture for (& - e)-free graphs from first principles. The proof directly yields an o(pn*) algorithm for p-coloring a perfect (K4 - e)-free graph. 0 1987 Academic Press, Inc.

1. INTRODUCTION

A graph G is perfect if for all vertex-induced subgraphs H of G (including G itself), the chromatic number of H equals the size of the largest of H. The Strong Perfect Graph Conjecture (SPGC, for short), due to Berge Cll, says

SPGC: A graph is perfect is and only if it contains no hole or antihole.

A hole is a chordless odd-length (35) circuit and an antihole is the complementary graph of a hole. Any odd-length circuit whose vertices induce a subgraph that contains no triangle must contain a hole. For more information about perfect graphs, the reader is referred to the books by Golumbic [3] and Berge and Chvatal [2]. This paper verities the SPGC for (K4 - e)-free graphs, that is, graphs whose edgesare each in just one (maximal) clique. Figure 1 shows a K4 - e subgraph. Our proof will start from first principles and will be constructive, yielding an O(pn2) algorithm to p-color an n-vertex hole-free (K4 - e)-free graph with maximum clique size p. Parthasarathy and Ravindra presented an incorrect proof of the SPGC for (K4 - e)-free graphs [6, on page 99 line - 8, the assertion “WV, + 1 4 E’ is false]. Their proof also was not from first principles; it used an important

* This research was partially supported by the National Science Foundation, grant DMS 8301934 313 0095-8956/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved. 314 ALANTUCKER

FIGURE 1 result by Lovasz [5] about critical perfect graphs (graphs that are not per- fect but all of whose proper vertex-induced subgraphs are perfect). Since the chromatic number of perfect graphs can be found in polynomial-time (see [4]), it is desirable to have constructive proofs of the SPGC that yield polynomial-time coloring algorithms. All graphs G = (V, E) in this note will consist of a finite vertex set V and a set E = { vivj} of undirected edges. A clique is a set of mutually adjacent vertices and is set-theoretically maximal (a clique cannot be properly con- tained a larger clique). Given a path P and distinct vertices x, y on P, let (x, P, y) denote the segment of P going from x to y. A graph is (properly) k-colored if each vertex is assigned one of k colors such that adjacent vertices receive different colors. If G has been colored, we can form the subgraph G, of G generated by all vertices of colors i and j. An i-j interchange at x interchanges the names of colors i and j in the component of G, containing x. In trying to color hole-free graphs, we can nssurnethat every clique has size at least 3: if an edge uu forms a 2-clique we remove it and after color- ing the graph reattach edge uv; if u and r both have color i, perform an i-j interchange at U, for some j, j# i (this interchange cannot affect v, or else there would be a chordless even-length i-j path from u to v that forms a hole with edge uu).

2. SPGC FOR (K,-e)-FREE GRAPHS

The critical theorem we use to color (& - e)-free graphs is:

THEOREM 1. Any hole-free (K, - e)-free graph with all cliques of size > 3 contains a vertex belonging to at most two cliques.

Proof This proof starts like the argument of Parthasarathy and Ravindra in [6] by building a chordless path in a hole-free (K4 - e)-free graph G, but the length and purpose of our path is different. COLORING PERFECT (& - e)-FREE GRAPHS 315

FIGURE 2

Starting from any given vertex or, we build a chordiess path p= (Ul, u2, Us,...), where edge zliui+ r is in the (unique) clique Ci, such that for i + 1 , 3, and so C’ - vk contains two distinct vertices w’, 1~“. Let v,w’ and u, w” be edges. Then (v,, P, u,.) must be even-length (since I(u,) and I(v,) are both odd). Hence (w’, v,, P, v,, w”, w’) is an odd-length (35) circuit free of triangular chords and contains a hole. This contradiction proves that any clique C’ containing uk (C’ # Ck ~ r ) must contain at least one vertex w that is in some Ci and forms the triangle ( W, Vi, vi+ r), but no other triangles with P (w cannot be in two or more Cj by the definition of P). Suppose vk is a member of three cliques: C, ~, and two other cliques D, and D,. Let wr and w2 be vertices in D, and D,, respectively, that form triangles with v,., u, + r and v,, us+ r , respectively (i.e., w1 E C, and w2 E C,). See Fig. 3. Since G is (K4 - e)-free, w, w2 $ E. Assume

P

FIGURE 3 316 ALAN TUCKER

Y

COROLLARY. An n-vertex, hole-free, (K4 - e)-free graph G with maximal clique size p and no 2-cliques has < 2pn edges. Proof. Proof by induction. Let x be in two cliques of G, as in Theorem 1. Then deg(x) < 2(p - 1). If either (or both) of x’s cliques were a triangle such as {x, a, b}, then G-x will have the 2-clique (a, h). In this case, also remove the edge (a, b) from G-x; as noted earlier, removing a 2-clique will not create a hole. By induction, the resulting (n - 1)-vertex graph G-x which is hole-free, (K4 - e)-free and 2-clique-free has <2p(n - 1) edges. Thus G has less than 2p(n - 1) + [2(p - 1) + 2]= 2pn edges. 1 Theorem 1 yields a simple induction proof of our key result.

THEOREM 2. The Strong Perfect Graph Conjecture is true for (K4 -e)- free graphs: for (K, - e)-f ree graphs, being hole-free implies perfection. ProoJ Our proof is by induction on the number 12of vertices in the graph. It is trivial for n = 1. Suppose that any (n - 1)-vertex hole-free (K4 - e)-free graph is perfect, and let G be such an n-vertex graph. By induction, any proper H of G is perfect. To prove that G is perfect, it remains to show that G can be p-colored, where p is the maximum clique size in G. As noted at the end of Section 1, we can assume initially that our graph G (and G - t)i - v2, etc.) have no 2-cliques and then insert them later. By induction G - v can be p-colored for any vertex u. By Theorem 1, choose u to be a vertex in two cliques of G, K,, and K,. (If v is in just one clique, it is trivial to color v properly.) Since there are p colors but 1K, - VI

THEOREM 3. An n-vertex perfect (K4 - e)-free graph G can be p-colored in O(pn2) steps, where p is the maximal clique size. Here we have assumed the clique-vertex incidence matrix was given. This incidence matrix takes O(n3) steps to construct from the adjacency matrix: the vertices adjacent to vertex x, can be partitioned in O(n2) steps into disjoint cliques using the fact that an edge is in only one clique: then delete .‘cI and all edges in these cliques and repeat the process for G - x1, etc.

REFERENCES

1. C. BEKGE, “Farbung von Graphen, deren Samtliche bzw deren ungerade Kreise Starr sind,” Wiss. z. Martin-Luther-Unix, Halle- Wittenberg Math.-Natur. Reihe (1961), 114-l 15. 2. C. BERGE AND V. CHVATAL. “Perfect Graphs,” North-Holland-Elsevier, New York, 1985. 3. M. GOLUMBIC, “Perfect Graphs and Algorithmic ,” Academic Press, New York, 1980. 4. M. GROTSCHEL,L. LOVASZ. AND A. SCHRIJVER,The and its consequences in combinatorial optimization, Con~binatorica 1 (191 l), 169-197. 5. L. LOVASZ, A characterization of perfect graphs, J. Combin. Theory Ser. B 13 (1972), 95-98. 6. K. PARTHASARATHYAND G. RAWNDRA, The validity of the strong perfect graph conjecture for (K,-e)-free graphs, J. Combin. Theory Ser. B 26 (1979) 98-100.